Download or read book Matrix Based Multigrid written by Yair Shapira and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many important problems in applied science and engineering, such as the Navier Stokes equations in fluid dynamics, the primitive equations in global climate mod eling, the strain-stress equations in mechanics, the neutron diffusion equations in nuclear engineering, and MRIICT medical simulations, involve complicated sys tems of nonlinear partial differential equations. When discretized, such problems produce extremely large, nonlinear systems of equations, whose numerical solution is prohibitively costly in terms of time and storage. High-performance (parallel) computers and efficient (parallelizable) algorithms are clearly necessary. Three classical approaches to the solution of such systems are: Newton's method, Preconditioned Conjugate Gradients (and related Krylov-space acceleration tech niques), and multigrid methods. The first two approaches require the solution of large sparse linear systems at every iteration, which are themselves often solved by multigrid methods. Developing robust and efficient multigrid algorithms is thus of great importance. The original multigrid algorithm was developed for the Poisson equation in a square, discretized by finite differences on a uniform grid. For this model problem, multigrid exhibits extremely rapid convergence, and actually solves the problem in the minimal possible time. The original algorithm uses rediscretization of the partial differential equation (POE) on each grid in the hierarchy of coarse grids that are used. However, this approach would not work for more complicated problems, such as problems on complicated domains and nonuniform grids, problems with variable coefficients, and non symmetric and indefinite equations. In these cases, matrix-based multi grid methods are in order.
Download or read book Matrix Based Multigrid written by Yair Shapira and published by Springer. This book was released on 2008-11-01 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Matrix-Based Multigrid introduces and analyzes the multigrid approach for the numerical solution of large sparse linear systems arising from the discretization of elliptic partial differential equations. Special attention is given to the powerful matrix-based-multigrid approach, which is particularly useful for problems with variable coefficients and nonsymmetric and indefinite problems. This book can be used as a textbook in courses in numerical analysis, numerical linear algebra, and numerical PDEs at the advanced undergraduate and graduate levels in computer science, math, and applied math departments. The theory is written in simple algebraic terms and therefore requires preliminary knowledge only in basic linear algebra and calculus.
Download or read book Multilevel Block Factorization Preconditioners written by Panayot S. Vassilevski and published by Springer Science & Business Media. This book was released on 2008-10-22 with total page 527 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is the first to provide a comprehensive, self-contained and rigorous presentation of some of the most powerful preconditioning methods for solving finite element equations in a common block-matrix factorization framework. The book covers both algorithms and analysis using a common block-matrix factorization approach which emphasizes its unique feature. Topics covered include the classical incomplete block-factorization preconditioners, the most efficient methods such as the multigrid, algebraic multigrid, and domain decomposition. This text can serve as an indispensable reference for researchers, graduate students, and practitioners. It can also be used as a supplementary text for a topics course in preconditioning and/or multigrid methods at the graduate level.
Download or read book Numerical Solution of Partial Differential Equations on Parallel Computers written by Are Magnus Bruaset and published by Springer Science & Business Media. This book was released on 2006-03-05 with total page 491 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the dawn of computing, the quest for a better understanding of Nature has been a driving force for technological development. Groundbreaking achievements by great scientists have paved the way from the abacus to the supercomputing power of today. When trying to replicate Nature in the computer’s silicon test tube, there is need for precise and computable process descriptions. The scienti?c ?elds of Ma- ematics and Physics provide a powerful vehicle for such descriptions in terms of Partial Differential Equations (PDEs). Formulated as such equations, physical laws can become subject to computational and analytical studies. In the computational setting, the equations can be discreti ed for ef?cient solution on a computer, leading to valuable tools for simulation of natural and man-made processes. Numerical so- tion of PDE-based mathematical models has been an important research topic over centuries, and will remain so for centuries to come. In the context of computer-based simulations, the quality of the computed results is directly connected to the model’s complexity and the number of data points used for the computations. Therefore, computational scientists tend to ?ll even the largest and most powerful computers they can get access to, either by increasing the si e of the data sets, or by introducing new model terms that make the simulations more realistic, or a combination of both. Today, many important simulation problems can not be solved by one single computer, but calls for parallel computing.
Download or read book A Multigrid Tutorial written by William L. Briggs and published by SIAM. This book was released on 2000-07-01 with total page 318 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- Numerical Analysis.
Download or read book multigrid methods written by Stephen F. Mccormick and published by CRC Press. This book was released on 2020-08-11 with total page 665 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collection of research papers on a wide variety of multigrid topics, including applications, computation and theory. It represents proceedings of the Third Copper Mountain Conference on Multigrid Methods, which was held at Copper Mountain, Colorado.
Download or read book Multigrid Methods written by Ulrich Trottenberg and published by Academic Press. This book was released on 2001 with total page 652 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- Numerical Analysis.
Download or read book Hierarchical Matrix Based Smoother for the Multigrid Method written by David Priebel and published by . This book was released on 2007 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Multigrid Methods III written by HACKBUSCH and published by Birkhäuser. This book was released on 2013-11-22 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: These proceedings contain a selection of papers presented at the Third European Conference on Multigrid Methods which was held in Bonn on October 1-4, 1990. Following conferences in 1981 and 1985, a platform for the presentation of new Multigrid results was provided for a third time. Multigrid methods no longer have problems being accepted by numerical analysts and users of numerical methods; on the contrary, they have been further developed in such a successful way that they have penetrated a variety of new fields of application. The high number of 154 participants from 18 countries and 76 presented papers show the need to continue the series of the European Multigrid Conferences. The papers of this volume give a survey on the current Multigrid situation; in particular, they correspond to those fields where new developments can be observed. For example, se veral papers study the appropriate treatment of time dependent problems. Improvements can also be noticed in the Multigrid approach for semiconductor equations. The field of parallel Multigrid variants, having been started at the second European Multigrid Conference, is now at the centre of interest.
Download or read book Matrix Computations written by Gene H. Golub and published by JHU Press. This book was released on 2013-02-15 with total page 781 pages. Available in PDF, EPUB and Kindle. Book excerpt: This revised edition provides the mathematical background and algorithmic skills required for the production of numerical software. It includes rewritten and clarified proofs and derivations, as well as new topics such as Arnoldi iteration, and domain decomposition methods.
Download or read book The Robust Multigrid Technique written by Sergey I. Martynenko and published by Walter de Gruyter GmbH & Co KG. This book was released on 2017-09-25 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a detailed description of a robust pseudomultigrid algorithm for solving (initial-)boundary value problems on structured grids in a black-box manner. To overcome the problem of robustness, the presented Robust Multigrid Technique (RMT) is based on the application of the essential multigrid principle in a single grid algorithm. It results in an extremely simple, very robust and highly parallel solver with close-to-optimal algorithmic complexity and the least number of problem-dependent components. Topics covered include an introduction to the mathematical principles of multigrid methods, a detailed description of RMT, results of convergence analysis and complexity, possible expansion on unstructured grids, numerical experiments and a brief description of multigrid software, parallel RMT and estimations of speed-up and efficiency of the parallel multigrid algorithms, and finally applications of RMT for the numerical solution of the incompressible Navier Stokes equations. Potential readers are graduate students and researchers working in applied and numerical mathematics as well as multigrid practitioners and software programmers. Contents Introduction to multigrid Robust multigrid technique Parallel multigrid methods Applications of multigrid methods in computational fluid dynamics
Download or read book Scientific Computing in Electrical Engineering written by Andreas Bartel and published by Springer. This book was released on 2016-05-09 with total page 251 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collection of selected papers presented at the 10th International Conference on Scientific Computing in Electrical Engineering (SCEE), held in Wuppertal, Germany in 2014. The book is divided into five parts, reflecting the main directions of SCEE 2014: 1. Device Modeling, Electric Circuits and Simulation, 2. Computational Electromagnetics, 3. Coupled Problems, 4. Model Order Reduction, and 5. Uncertainty Quantification. Each part starts with a general introduction followed by the actual papers. The aim of the SCEE 2014 conference was to bring together scientists from academia and industry, mathematicians, electrical engineers, computer scientists, and physicists, with the goal of fostering intensive discussions on industrially relevant mathematical problems, with an emphasis on the modeling and numerical simulation of electronic circuits and devices, electromagnetic fields, and coupled problems. The methodological focus was on model order reduction and uncertainty quantification. this book="" will="" appeal="" to="" mathematicians="" and="" electrical="" engineers.="" it="" offers="" a="" valuable="" starting="" point="" for="" developers="" of="" algorithms="" programs="" who="" want="" learn="" about="" recent="" advances="" in="" other="" fields="" as="" well="" open="" problems="" coming="" from="" industry.="" moreover,="" be="" use="" representatives="" industry="" with="" an="" interest="" new="" program="" tools="" mathematical="" methods.
Download or read book Challenges in Scientific Computing CISC 2002 written by Eberhard Baensch and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: The conference Challenges In Scientific Computing (CISC 2002) took place from October, 2 to 5, 2002. The hosting institution was the Weierstrass Insti tute for Applied Analysis and Stochastics (WIAS) in Berlin, Germany. The main purpose of this meeting was to draw together researchers working in the fields of numerical analysis and scientific computing with a common interest in the numerical treatment and the computational solution of systems of nonlinear partial differential equations arising from applications of physical and engineering problems. The main focus of the conference was on the problem class of non linear transport/diffusion/reaction systems, chief amongst these being: the Navier-Stokes equations, semiconductor-device equations and porous media flow problems. The emphasis was on unsolved problems, challenging open questions from applications and assessing the various numerical methods used to handle them, rather than concentrate on accurate results from "solved" problems. Thanks to the participants it was an interesting meeting. The presentations stimulated exchanging ideas and lively discussions. This proceedings comprises 13 papers form the conference, ranging from numerical methods for flow problems, multigrid methods, semiconductor and microwave simulation, solution methods, finite element analysis to software aspects. This interesting conference would not have been possible without the help of the staff of the WIAS. I thank all participants, and all our supporters, especially those not onstage, for making the conference a success.
Download or read book Image Processing Based on Partial Differential Equations written by Xue-Cheng Tai and published by Springer Science & Business Media. This book was released on 2006-11-22 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book publishes a collection of original scientific research articles that address the state-of-art in using partial differential equations for image and signal processing. Coverage includes: level set methods for image segmentation and construction, denoising techniques, digital image inpainting, image dejittering, image registration, and fast numerical algorithms for solving these problems.
Download or read book Analysis of an Aggregation based Algebraic Multigrid Method and Its Parallelization written by Meng-Huo Chen and published by . This book was released on 2014 with total page 92 pages. Available in PDF, EPUB and Kindle. Book excerpt: The interests of this thesis are twofold. First, a two-grid convergence analysis based on the paper [Algebraic analysis of aggregation-based multigrid by A. Napov and Y. Notay, Numer. Lin. Alg. Appl. 18 (2011), pp. 539-564 ] is derived for various aggregation schemes applied to a finite element discretization of a rotated anisotropic diffusion equation. As expected, it is shown that the best aggregation scheme is one in which aggregates are aligned with the anisotropy. In practice, however, this is not what automatic aggregation procedures do. We suggest an approach for determining appropriate aggregates based on eigenvectors associated with small eigenvalues of a block splitting matrix. In the second part of the thesis several issues regarding the parallel implementation of aggregation-based multigrid methods are discussed. The coarsest grid solving stage of multigrid cycles has been a bottleneck for parallel multigrid algorithms to attain a good speedup. A comparison between a parallel linear system direct solver (MUMPS) and a few steps of preconditioned conjugate gradient (PCG) methods for solving the coarsest grid system is carried out and tested on TACC Lonestar multi-processor machine. Regarding the preconditioner of conjugate gradient iterations, a parallel sparse approximate inverse (SAI) algorithm is used to construct an approximate inverse of the original matrix in order to replace the preconditioner solving step, which is inherently sequential, by matrix-vector multiplications. The linear systems tested arise from discretization of 2D or 3D partial differential equations, which are symmetric positive definite. The results exhibit that using PCG on the coarsest grid attains better speedup and overall better performance than MUMPS when the number of processors is greater than about 100. The effects of different decompositions of the physical domain (rows/slab versus blocks/pencils) on the scaling and efficiency of aggregation-based algebraic multigrid are also studied and one sees that the blocks/pencils decomposition of the physical domain reduces the amount of communication and hence has better performance.
Download or read book Iterative Methods for Sparse Linear Systems written by Yousef Saad and published by SIAM. This book was released on 2003-04-01 with total page 537 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- General.
Download or read book Algebraic Multigrid for the Multi ion Transport and Reaction Model written by Peter Thum and published by Logos Verlag Berlin GmbH. This book was released on 2012 with total page 229 pages. Available in PDF, EPUB and Kindle. Book excerpt: Das Multi-Ionen Transport und Reaktionsmodell wird fur die Simulation von elektrochemischen Prozessen eingesetzt. Das durch das Modell gegebene System partieller Differentialgleichungen (PDE) wird mit Hilfe einer gemischten Residuen-Distribution und Finiten Elemente Methode diskretisiert und mit dem Newton Verfahren linearisiert. Dabei entstehen eine Reihe linearer Gleichungssyteme.Die Dissertation beschreibt ein physikalisch orientiertes algebraisches Mehrgitterverfahren, welches zur effizienten und robusten Losung dieser linearen Gleichungssyteme eingesetzt werden kann. Insbesondere wird auf die Reihenfolge der Variablen und deren Wirkung auf das Glattungsverhalten eingegangen. Bei der Konstruktion der Grobgitterkorrektur werden Aspekte wie eine verletzte Peclet Bedingung und die Nichtlinearitat des PDE Systems beachtet.
